Question

У класі 35 учнів. 24 з них відвідують театральний гурток,12- танцювальний гурток, а 8 учнів не беруть участі в роботі цих гуртків. Скільки учнів відвідують обидва гуртки?

Answer 1

35 - 8 = 27 учеников заняты в кружках
27 - 24 = 3 ученика участвуют только в танцах
27 - 12 = 15 учеников участвуют только в театре
15 + 3 = 18 учеников участ. только в одном кружке
27 - 18 = 9 учеников участ. в двух кружках
Ответ 9 учеников.

Answer 2

I can answer your question in Ukrainian. The user asked me to answer in detail how many students attend both clubs if there are 35 students in the class, 24 of them attend the theater club, 12 the dance club, and 8 students do not participate in the work of these clubs.

The answer is 9 students. Here is how to find it:

- First, subtract the number of students who do not participate in any club from the total number of students in the class. This gives the number of students who participate in at least one club. $$35 - 8 = 27$$ - Second, subtract the number of students who attend the theater club from the number of students who participate in at least one club. This gives the number of students who participate only in the dance club. $$27 - 24 = 3$$ - Third, subtract the number of students who attend the dance club from the number of students who participate in at least one club. This gives the number of students who participate only in the theater club. $$27 - 12 = 15$$ - Fourth, add the number of students who participate only in the dance club and the number of students who participate only in the theater club. This gives the number of students who participate in exactly one club. $$3 + 15 = 18$$ - Fifth, subtract the number of students who participate in exactly one club from the number of students who participate in at least one club. This gives the number of students who participate in both clubs. $$27 - 18 = 9$$

You can also find the answer by using a Venn diagram. A Venn diagram is a graphical way of representing the relationships between sets of objects. In this case, the sets are the students who attend the theater club and the students who attend the dance club. The diagram looks like this:

    T = theater club
    D = dance club
    U = universal set (all students in the class)

U = {1, 2, 3, ..., 35}
    T = {1, 2, 3, ..., 24}
    D = {9, 10, 11, ..., 20}

Venn diagram:

|-----------------U-----------------|
    |                                   |
    |        |--------------|           |
    |        |      T       |           |
    |        |              |           |
    |        |   |------|   |           |
    |        |   |   D  |   |           |
    |        |   |      |   |           |
    |        |   |------|   |           |
    |        |              |           |
    |        |--------------|           |
    |                                   |
    |-----------------------------------|

The region where the two circles overlap represents the students who attend both clubs. To find the number of students in this region, we can use the formula:

$$|T \cap D| = |T| + |D| - |T \cup D|$$

where |A| means the number of elements in the set A, T ∩ D means the intersection of the sets T and D (the elements that belong to both sets), and T ∪ D means the union of the sets T and D (the elements that belong to either set or both).

Plugging in the numbers, we get:

$$|T \cap D| = 24 + 12 - 27 = 9$$

This matches the answer we found by using subtraction.

I hope this helps you understand the problem and the solution. If you have any more questions, feel free to ask me.